Attributing changes in extreme daily precipitation to global warming is difficult, even when based on global climate model simulations or statistical trend analyses. The question about trends in extreme precipitation and their causes has been elusive because of climate models' limited precision and the fact that extremes are both rare and occur at irregular intervals. Here a newly discovered empirical relationship between the wet-day mean and percentiles in 24 h precipitation amounts was used to show that trends in the wet-day 95th percentiles worldwide have been influenced by the global mean temperature, consistent with an accelerated hydrological cycle caused by a global warming. A multiple regression analysis was used as a basis for an attribution analysis by matching temporal variability in precipitation statistics with the global mean temperature.
 Changes in extreme rainfall statistics can have severe consequences for societies and ecosystems, which makes the analysis of precipitation statistics more than just an academic question [Field et al., 2012]. The body of scientific publications suggests that rainfall patterns will change as a result of a global warming, for which more heavy precipitation events are expected [Solomon et al., 2007; Tebaldi et al., 2006; Chou and Lan, 2012; Durack et al., 2012; Giorgi et al., 2011; Peterson et al., 2012; Trenberth, 2011; Christensen and Christensen, 2002; Groisman et al., 2005]; however, a key message from the 2012 IPCC SREX report was “[t]here is medium confidence that anthropogenic influences have contributed to intensification of extreme precipitation at the global scale” [Field et al., 2012]. This conclusion was based on trend analysis of rain gauge data, model results, the expectation that an increase in the atmospheric moisture content will lead to an increase in extreme precipitation, and a detectable anthropogenic influence on atmospheric moisture content. From the Clausius-Clapeyron relation, higher temperatures imply higher saturation vapor pressure (∼7%/°C), and hence greater capacity for air to hold moisture [Willett et al., 2010; Trenberth et al., 2003]. More recent studies also provide further support for a connection between the global hydrological cycle and global warming [Min et al., 2011; Chou and Lan, 2012; Durack et al., 2012; Giorgi et al., 2011; O'Gorman, 2012; Berg et al., 2013].
 The objective of this analysis is to provide a firmer answer to the question whether the recent global warming [Solomon et al., 2007] has influenced the extreme precipitation statistics through new and independent analyses. The working hypothesis is that if the hydrological cycle responds to the global mean temperature (T), then this implies the possibility to predict some aspects of precipitation with statistical models. The influence from T on local precipitation, however, is expected to emerge on long time scales rather than on a daily basis. If such a link exists, it is postured that it can be utilized for downscaling precipitation statistics.
 Recent studies have demonstrated that heavy precipitation amounts, such as the wet-day 95th percentile (q95), are functions of the wet-day mean (μ) and the wet-day frequency (fw), derived from spatial variations in the daily precipitation statistics [Benestad et al., 2012a]. This link was extended here to provide a means for attributing trends in extreme precipitation to global warming through the means of regression analyses for identifying temporal fingerprints associated with T [Min et al., 2011; Stott et al., 2001].
2 Methods and Data
 There are two aspects of precipitation which can be represented by different terms: (1) whether it rains or not, and (2) how much water precipitate out when it rains. These two factors may respond to different conditions, and hence an analysis of precipitation should account for these differences and involve the two variables fw and μ. Here the emphasis was put on wet-day mean μ rather than frequency because previous analysis has indicated that the wet-day percentiles were much more sensitive to μ [Benestad et al., 2012a]. Future work will address dependencies for fw.
Benestad et al. [2012a, 2012b] found that 24 h precipitation tends to have a statistical distribution that is approximately similar to an exponential distribution, albeit with a heavier upper tail. Here it is hypothesized that a simple exponential distribution can provide a crude description of temporal changes in the 24 h precipitation, as suggested by Benestad . For exponentially distributed values, it can be shown that any percentile is proportional to the mean μ [Benestad et al., 2012b]:
 In this case, an estimate of μ can provide information about higher percentiles. Furthermore, the mean estimate tends to be less resistant to outliers than high percentiles, and its estimate has a standard error that diminishes with the sample size n according to . Another motivation for studying μ rather than the percentiles qp is that μ is believed to be easier to downscale than high percentiles because it is less sensitive to statistical fluctuations. Analyses of high percentiles require samples of sufficient size for robust estimates of the higher percentiles, and hence the data were sorted into 5 year segments (pentads), from which the wet-day mean and percentiles were estimated. The pentads emphasized moderately long time scales (5 years), and corresponding pentads were used to estimate mean values of T. The analysis was repeated with 8 year and 10 year segments, although the analysis for longer sequences was limited due to reduced number of data points in time available for regression analysis (both in terms of rain gauge records, since longer periods were needed to get a minimum number of sequences, as well as number of sequences for each of these).
 The notation used henceforth is μ to represent the wet-day mean in general, and μ(x,t) is used to refer to the value of μ at particular location x and the pentad representing time t. The observed wet-day mean will be referred to as μ whereas the corresponding predicted value will be represented by , and model prediction based on T is . The analysis was applied to μ(x,t)/μref(x) where the reference value μref(x) was taken to be the mean value over the calibration period: . Here nt is the number of pentads over which the sum was based. In order to get a maximum number of stations with complete data coverage for the calibration period, the time period was restricted to 1945–1995, as some data records end earlier than others. The detection of the dependency to the global mean temperature fails for short calibration periods (e.g., 1945–1990, nt≤10).
 The analysis was made over both temporal and spatial domains, and the geographical and temporal variations in precipitation were expressed mathematically as data matrix X. This matrix had two dimensions where each column contained the number of sites nx (space dimension) and each row the number of pentads nt (time dimension). (There were nt different columns all of which had the length nx.) The elements in the matrix X can be written as xx,t and contained μ(x,t)/μref(x) or q95(x,t)/μref(x).
 A principal component analysis (PCA) was applied to data matrix X containing the pentad values in order to make the calculations efficient and place emphasis on the large-scale features. Here a singular value decomposition [Strang, 1988; Press et al., 1989] was used:
 The matrix U contained the empirical orthogonal functions (EOFs) [Lorenz, 1959] Λ the eigenvalues, and VT is the transpose of the principal components. U described the spatial precipitation patterns, while V contained the temporal variations in the pentad values. Here the notation vi(t) refers to the ith principal component and t is used to emphasize that V represents the variations in time. Likewise, the notation ui(x) is used to refer to one EOF describing a spatial precipitation pattern. The left and right eigenvectors Uand V have the property of being orthogonal, which makes the estimation of the regression coefficients less prone to spurious influence from other covariates [Benestad and Schmidt, 2009].
 A multiple linear regression analysis was used to test the hypothesized influence from T on μ and provided a basis for the attribution analysis. The link to the global mean temperature has been used in other trend studies [Oldenborgh et al., 2013, 2009], and the analysis carried out here followed a similar approach. To improve the signal-to-noise ratio, well-known interannual variations associated with El Niño Southern Oscillation (ENSO) were filtered out by including an additional term N in the regression equation to describe its influence in a similar way as in Foster and Rahmstorf .
 The regression analysis was applied to detrended covariates to avoid spurious influence from coinciding long-term trends, and an extra term t was included to account for the trends. The symbol T′ is used to represent the detrended temperature according to T(t)=T0+cTt+T′, where the notation T(t) is used to emphasize the complete temporal evolution in the global mean temperature. The symbol cT describes the best-fit linear trend in the global mean temperature and T0 is a constant, leaving the detrended temperature T′ (residual from regression against time t) with zero mean and zero linear trend. Likewise, N′ is the detrended NINO3.4 index (N=N0+cNt+N′). The reason for detrending T and N and adding a separate trend term t in the regression equations was that a regression analysis is less likely to attribute spurious weights to T in the absence of colinearity such as common long-term trends [Benestad and Schmidt, 2009].
 Generalized linear models (GLMs) [Dobson, 1990; McCullagh and Nelder, 1989] were used to estimate dependencies for μ/μref and q95/μref to the covariates. The model fitting involved iteratively re-weighted least squares, assuming a Gaussian noise distribution for μ/μref and q95/μref with an identity link function. The analysis was also repeated with ordinary linear regression (OLR; least squares), and the results were not sensitive to the choice of regression strategy. The regression was applied to each of the i principal components vi in order to estimate the best-fit coefficients describing a linear dependency to the covariates:
 The β's in equation (3) are the regression coefficients, describing the degree of dependency between vi(t) and the covariates. They may be regarded as weighting functions which together provide weighted combination of time, global mean temperature and ENSO that best matches the time variations in vi(t). The temporal trends in μ/μref were then computed based on the matrix product between the PCA components (equation (2)) where the principal component was replaced by the regression coefficients on the right hand side in equation (3). We can introduce a new variable αk, where subscript k refers to any of the terms t, N′, or T′ in equation (3), and take the sum of the matrix product for each PCA mode according to:
 With this substitution, we can rewrite equation (3) in terms of an expanded mathematical notation of X for which we explicitly take the values xx,t to be μ:
 This equation provides a transparent framework for the subsequent analysis, where the coefficients αt represent estimates for long-term linear trends and t is the time in years. The term ζ represents unexplained variability (noise) and αT describes the dependency to T. The trends in 95th percentile were estimated similarly although with a simpler expression
 Here the terms with T′ and N′ (which are detrended) have been dropped since the objective was only to estimate the linear trend in q95. The calibration interval was 1945–1995 (the period with most rain gauges in operation), comprised 11 pentads, and included 7109 rain gauge records for 24 h wet-day precipitation amount from the Global Daily Climatology Network (GDCN) [Peterson et al., 1997; Legates and Willmott, 1990a, 1990b; Lanzante, 1996] and the data archive of the Norwegian Meteorological Institute. The global mean temperature was calculated from the 2°×2° gridded annual mean GISTEMP (1200 km) [Hansen and NASA/GISS, 2009] and the NINO3.4 was taken from the IRI/LDEO Climate Data Library (Columbia University; (http://iridl.ldeo.columbia.edu/SOURCES/.Indices/.nino/.EXTENDED/.NINO34/gridtable.tsv).
 The regression model, calibrated over the 1945–1995 interval, was subsequently used to predict pentad values in order to evaluate against independent data. The predictions can be expressed mathematically as . To predict the long-term trends, the detrended temperature T′ used for calibrating the model was replaced with the original temperature T, and the time term αt and ENSO term αN were set to zero. The combination of PCA products and T made it possible to predict values for sites and pentads for which rain gauge records had missing data, but these were masked in the evaluation.
 The analysis was done in the R environment [R Development Core Team, 2004; Gentleman and Ihaka, 2000], and the source code used to carry out all the analyses and figures is available through the R-package Benestad-JGR2013 (version 1.10) provided in the supporting information (The tar.gz-version was compiled for Linux/Mac whereas the zip-version was made for Windows platforms). This package provides the open source R scripts, a documentation of the code, and the necessary data for replicating the analysis presented here. The R environment is free and platform-independent, and the R-package provided in the supporting information can be used on any PC with R installed (freely available together with complete documentation and manuals from http://cran.r-project.org).
 Table 1 provides a listing of the main statistics on the regression carried out according to equation (3), and the results suggested statistical significant results (at the 10% level) for βt(i) and βT(i) only when both scored high and only for the three leading EOFs (i∈1,3). Since the correlation between t and T′ was zero (not shown), the probability for seeing this match by accident was small: the probability that three βt(i)s have p value <0.1 and the probability that three βT(i)s have p value <0.1, and the probability that the same three modes i gave significant results for βt(i) and βT(i) from all possible 3-of-11 combinations. This probability was estimated from binomial distribution for the number of significant results for both coefficients (; p=0.1; ), and the number of combinations for which three significant results can be attributed to 11 different modes (165 combinations, one of which matches the other coefficient). The p value is . Furthermore, the fact that all the significant results were for leading EOFs means that they involved only the large-scale spatially coherent modes. It is hypothesized here that the global temperature affects the precipitation statistics on the global rather than local scales and over long rather than short time scales.
Table 1. ANOVA for the Regression Against the PCs Based on an Ordinary Linear Regressiona
P Value All
P Value βT
P Value βt
Note that the good fit to T′ and t were found for the same modes, although the results for βt(2) was only significant at the 10% level. The values shown in bold above were significant at the 5% level.
 Figure 1 shows a comparison between trends in μ and q95, using expression (1) to estimate values for μ that are comparable to q95 and the values shown along the axes are −ln(1−0.95)μ/μref and q95/μref, respectively. For the vast majority of the points, there was a close one-to-one correspondence between the observed trends in q95 and those estimated from μ. In other words, realistic trend magnitudes in the upper percentiles can be calculated from estimated trends in the wet-day mean μ, assuming an exponential distribution. However, for some locations (their locations are marked in the inserted map, and are mainly located in Russia), the μ dependency gave too high trend estimates for q95. The correlation between the linear trends based on q95 and − ln(1−0.95)μ was 0.64, and the pvalue <2.2×10−16 assuming 7107 degrees of freedom (number of stations - 2; see the supporting information for further discussion about sample size and degrees of freedom).
 Figure 2 shows a comparison between 1945–1995 linear trends in μ and trends associated with the global mean temperature cTαT, where cT=dT/dt is the linear average rate of change (the lower right insert shows the full data set whereas the main plot shows a close-up of the majority of the data points). Here a potential influence from ENSO was taken into account by subtracting the trend in NINO3.4: αt′=αt−αNcN. The correlation between the observed linear trends in μ and αTcT was high (0.71), although most points were scattered around origo (p value <2.2×10−16 assuming 7107 degrees of freedom). A number of points also scattered around a straight line; however, the slope was a little greater than unity. The location of these points are shown in the inserted map, and represents mainly Russian locations. In other words, the temperature dependency appeared to overestimate the trends in μ to some degree for the sites with the highest trend values, which may suggest that other factors also influence the trends in the precipitation statistics, such as landscape changes, effects from irrigation, changes in circulation regimes, or the local sea surface temperature. There may also be some data issues, and another explanation may be that the dependency was not linear.
 Figure 3 shows the geographical distribution of the 1945–1995 linear trends cTαT associated with T. There were mostly increases in μ over the northern hemisphere midlatitudes, with strongest response over Eurasia and southern states in the U.S. Some scattered locations with opposite trends were also found, especially in central parts of the U.S., Mexico, Brazil, and South Africa. The geographical clustering of the temperature dependencies may reflect real physical connections; however, isolated points with opposite signs also suggest that local fluctuations may have masked a global connection. For instance, potential local influences that could change the sign of the relationship between precipitation and temperature may include prevailing winds, orographic forcing, aerosols, irrigation, landscape changes, and regional sea surface temperature. Some of the isolated points of opposite trend result may also be associated with data quality issues. Previous work has found a tendency where regions with high average precipitation are those where the trends are the greatest (the “richer-get-richer” mechanism) [Chou et al., 2009; Liu et al., 2012]; however, those analyses focused on the rainfall totals (scale as the all-day mean) rather than the wet-day mean. There is a correlation between the mean precipitation and μ (supporting information) of 0.64; however, there are some regions with increases in μ which have low mean precipitation, most notably in California and over parts of Iberia, as well as most of Australia. In dry subtropical regions, the wet-day frequency fw is low, but once it rains, the amounts tend to be higher than other regions with more frequent precipitation. The physical mechanisms behind such rare and intense events may be convective storms [Berg et al., 2013] or “atmospheric rivers” [Ralph and Dettinger, 2011]. Changes in μ are not directly comparable to the precipitation totals or mean, since the wet-day frequency is an additional factor (). Other studies have found that the Northern Hemisphere midlatitudes are expected to get wetter [Zhang et al., 2007; Balan Sarojini et al., 2012]), i.e., higher , whereas the results here indicate a tendency of increasing amounts for those days when it rains, and higher wet-day q95.
 Figure 4 shows a comparison between observed μ(x,t)/μref (x) and predicted for all the different pentads and rain gauge locations. The picture is more noisy than for the trends, and is consistent with a pronounced presence of local processes influencing the precipitation statistics on time scales shorter than a decade. Grey points show results for the calibration period, but where detrended temperatures T′ have been replaced by the original values T. Hence, the temperature was used to predict both trend and pentad-to-pentad variations (both αt and αN were set to zero to emphasize the link to T). The correlation between the predicted and observed values was 0.59 (assuming 7109×11−2=78,197 degrees of freedom), and the high number of data points suggested that these were statistically significant at the 5% level. For the independent years, the correlation dropped to 0.22; however, the significance tests still suggested a low probability for chance correlation (p value <3.4×10−7 assuming 24,833 degrees of freedom). One interpretation of Figures 2 and 4 is that the global connection is seen on longer time scales than 5 years, and that local factors are more pronounced for time scales shorter than a decade. This finding is consistent with Table 1 which indicated significant regression results only for the three leading modes, representing the largest spatial and temporal scales. Other explanations may be that the number of rain gauge records with valid data was lower outside the calibration period (Figure S1), and the estimate of the global mean temperature before 1940 was based on a reduced global network of thermometers. Hence, the uncertainty level of the global mean T was greater for the early part of the record.
 Figure 5 shows the average of μ(x,t)/μref(x) over all the stations (thick solid black line) in addition to the 5th and 95th percentiles of the sample of stations (dashed lines). Hence, the curve provides a description of the aggregated statistics over large spatial scales (the locations are shown in Figure 3). It is important to note that these percentiles are different to q95; here the 5th and 95th percentiles represent the 90% interval of the values of μ(x,t)/μref(x) over all the stations. The 1945–1995 calibration period is marked by the grey line, and the black curves represent the independent data. The blue curve shows the corresponding averages for predicted values of based on the regression coefficients and the global mean temperature T. For this comparison, a mask was applied to ignore the locations and times of predictions where the corresponding rain gauges had missing values. The light blue part of the curve shows the calibration period. The station average trend in the predictions was similar to the trend in the corresponding observations, albeit slightly greater for the 1945–1995 period as indicated in Figure 2. The results also indicate that the trends estimated from the calibration against the detrended T′ gave a reasonable description of the long-term changes for the original T that was used as input. A comparison with the out-of-sample data (not used for the calibration) before 1945 and after 1995 indicates that the average value for the predictions described trends that diverges to those observed; however, the number of stations diminish severely outside this period. The explanation may be that the early record is subject to stronger sampling fluctuations and that it is less representative of a global aggregate.
 Local fluctuations carry less weight for the station average in Figure 5 (Table 1), which emphasizes the globally aggregated values, and the analysis suggests that there has been a tendency of increased μ that is consistent with the recent global warming. On the 5 year time scale, the two curves displayed some differences, suggesting skill only for longer time scales. Furthermore, the 90% confidence interval for the predicted was narrow compared to the observed range for the pentads within the calibration intervals, which again may suggest that the dependency to the global mean becomes more important for longer time scales and that there are other factors which have a more pronounced influence on the 24 h precipitation statistics on shorter time scales. The 1945–1995 average for μ(x,t)/μref(x) was 100% by definition, and the scatter in the pentad values over the calibration period was entirely due to pentad-to-pentad variability.
 The red curve shows the global mean T for comparison, although its scaling is arbitrary.
4 Discussion and Conclusions
 The analysis of the 24 h precipitation data indicated a clear connection between the global mean temperature and μ for time scales longer than a decade and for large-scale spatial aggregates. This connection can be extended with equation (1) and the results presented in Figure 1 to provide evidence for a systematic influence of T on the wet-day qp. The link between the wet-day percentiles and mean is also found over the spatial domain [Benestad et al., 2012b, 2012a]. These results therefore constitute a strong evidence that the global warming has caused more intense precipitation over the last century. However, these results were limited to regions with available 24 h precipitation data.
 The attribution analysis presented here involved a regression with respect to temporal fingerprints [Min et al., 2011] between the precipitation statistics and the global mean temperature, rather than spatiotemporal patterns found in model results and gridded observations [Stott et al., 2001]. This analysis provided an independent line of analysis to previous studies [Min et al., 2011; Tebaldi et al., 2006; Field et al., 2012], and the results confirmed the influence of T on μ proposed by Giorgi et al. ; Min et al. . Some differences to that of Min et al.  and other studies involved the use of different mathematical framework (e.g., Min et al. used extreme value distributions rather than the simpler exponential distribution), and using the bulk of the data to derive the trends rather than the upper tail. In addition, the present analysis made use of a “temporal fingerprinting” in terms of pentad-to-pentad variations, and an explicit distinction was made between wet and dry days. Min et al.  used a different data set for observations (the Hadley Centre global land-based gridded climate extremes data set), as opposed to individual station records (gridding may introduce spatial inhomogeneities in extreme values). Furthermore, they restricted their analysis to 1951–1999. In this respect, these results contributed toward the objective to provide a firmer answer to the question of whether the global warming has lead to more extreme 24 h precipitation.
 These results also suggest that downscaling of precipitation percentiles is feasible, as the trends in μ can be defined as a function of the global mean temperature, as with local temperature [Oldenborgh et al., 2009]. Such dependencies can be exploited to bypass problems pointed out by some critics concerning local representation of climate in general circulation models [Oreskes et al., 2010; Pielke Sr. and Wilby, 2012; Palmer, 2011]. Future work will involve the results reported here for downscaling of climate predictions and projections. However, it is necessary to carry out an additional analysis for the wet-day frequency fw in order to get a complete picture of the statistics [Benestad et al., 2012a]. The strategy presented here was appropriate for long-term and large-scale climate change scenarios, but the skill for regional and shorter 5 year and decadal time scales was limited as other conditions, such as the local (dew-point) temperature, appeared to be more important for the shorter terms [Lenderink et al., 2011; Berg et al., 2013]. A similar approach may nevertheless be appropriate for seasonal and decadal predictions if relevant regional large-scale conditions can be identified for shorter time scales and use these as predictors instead of the global mean temperature. Once connections between the seasonal wet-day mean precipitation and regional physical conditions have been established, it is then possible to downscale the probability density function (PDF) parameters through predictions such as and [Pryor et al., 2005a, 2005b; Pryor et al., 2006; Benestad, 2007]. The wet-day frequency fw∈0,1 may require another strategy than μ, e.g., GLMs assuming a binomial distribution and a logit-link function. A Bayesian approach can combine the downscaling of PDF parameters (e.g., μ) with ensembles of climate simulations.
 One key assumption for using this analysis to describe high percentiles is that the PDFs describing the statistical characteristics of the processes provide a signature of the underlying physics, and as long as the same underlying physical processes are present, although with different character, the PDF-family stays the same albeit with parameters shifted according to changed conditions. Hence, a connection between upper percentiles and the mean enables the attribution for rare and stochastic events because the upper tail of the PDF is expected to be less prone to sampling fluctuations than the actual data.
 Physics-based expectations of a link between T and precipitation further support for these results [Chou and Lan, 2012; Giorgi et al., 2011; Trenberth, 2011]. The physics connecting a global warming to the hydrological cycle involves both the large-scale circulation and evaporation near the surface. Higher T is expected to lead to higher atmospheric water vapor, and hence increase the mass of water available for precipitation [Held and Soden, 2000, 2006; Allen and Ingram, 2002], and the Clausius-Clapeyron relation implies that specific humidity would increase roughly exponentially with temperature [Allen and Ingram, 2002]. These expectations have been supported by empirical evidence, as Lenderink et al.  found a scaling relation between the local dew-point temperature TDP preceding the events and the hourly precipitation extremes in Hong Kong Observatory and the Netherlands to be 14%/TDP for TDP<23°C. Their results suggested a scaling twice as large as following from the Clausius-Clapeyron relation. Based on a comparison between model simulations and observations, Zhang et al.  concluded that the recent global warming may be associated with increased precipitation in the Northern Hemisphere midlatitudes, dryer conditions in the Northern Hemisphere subtropics and the Tropics, and moistening in the Southern Hemisphere subtropics and the deep Tropics, and Liu et al.  found a dependency between the global precipitation and temperature of ∼2%/K in climate model simulations and 34%/K in the observations. Over the tropical ocean, they found a tendency for wet regions to become wetter and dry regions drier with higher global mean temperature. The atmospheric large-scale circulation may change with higher global mean temperature, and Held and Soden  found that increased lower-tropospheric water vapor had profound consequences, such as a consistent decrease in convective mass fluxes in a range of different climate model simulations. They also found an increase in horizontal moisture transport and a strengthening of the patterns of evaporation-precipitation difference.
 Local conditions may affect the precipitation statistics through atmospheric moisture recycling [Trenberth, 1999], and the local moisture may influence the generation of convective storms [Taylor et al., 2012]. Soil moisture has also been associated with a feedback mechanisms for heat waves and droughts [Hirschi et al., 2011], and may hence have an influence on μ and fw. Berg et al.  have also suggested that there may be a change in type of precipitation with higher local temperatures, from stratiform to convective, which would be consistent with an increase in μ. For convective storms, the rain concentrates over smaller regions as showers, which may result in fewer rain events, but higher amounts where it rains. Changes in the atmospheric circulation in response to warming is expected to affect precipitation. One common denominator is the vertical energy flow from the surface to the heights where the heat escapes to space. A portion of this energy flow is carried as latent heat though convection, cloud formation, and condensation. Changes in the vertical energy flow will affect the hydrostatic stability of the atmosphere and cause vertical readjustments when different levels of the atmosphere experience different rates of heating. Changes in the atmospheric moisture (and mean lapse rate) and cloudiness also play a role for the vertical energy flow. A stronger greenhouse effect implies an increase in the optical depth that will perturb the vertical energy flow as well as an elevation of the height from which the planetary bulk heat loss takes place [Benestad, 2013].
 In conclusion, information based on PDFs for daily precipitation was utilized to attribute past trends in the high wet-day percentiles to the ongoing global warming. The dependency to the rising global mean temperatures was clear for long-term global aggregate, but other processes dominated on 5 year local scales. These results therefore confirmed the hypothesis that statistical links between T and precipitation become detectable on long time scales. The analysis proposed here may also lead to pioneering efforts in downscaling extreme precipitation.
 The work is an extension of the collaboration with NCAR and I want to acknowledge valuable discussions with Doug Nychka, Abdelkader Mezghani, and Arnoldo Frigressi. This work has been supported by the Norwegian Meteorological Institute and SPECS (EU grant agreement 3038378). I also acknowledge the Global Daily Climatology Network (GDCN) and their effort of collecting, analyzing, and archiving daily data into a single data set.